Adam LaRoche followed a grounder into right field, scoring Zimmerman with the tying run.
Ian Desmond (2-for-4, solo home run) followed by with a double into the left-center field gap, putting runners at second and third.
After Roger Bernadina was intentionally walked, Lombardozzi lifted a fly ball into left field and LaRoche came home with the walk-off game-winner.
The rally handed the decision to Tyler Clippard (4-1) after Nats starter Jordan Zimmermann had yielded just two unearned runs on four hits over eight innings.
New York's Jeremy Hefner was saddled with a no-decision after tossing seven innings of one-run ball, allowing four hits while striking out seven.
Omar Quintanilla provided a two-run triple for the Mets in the opener of a three-game series.